Chapter 2

Heat Transfer Calculations

Learning Objectives

At the end of this chapter students will:

• Be familiar with the mechanisms of heat transfer which are encountered in heat

exchangers and other engineering heat transfer problems

• Be able to calculate an overall heat transfer coefficient (or resistance) from

knowledge of individual heat transfer coefficients (or resistances)

• Understand the concept of fin efficiency and be able to determine overall heat

transfer coefficients in geometries involving extended surfaces.

Be able to apply published correlations to determine heat transfer coefficients in

single-phase, boiling and condensing flows. Heat transfer specialists may be able to

quote some correlations for convective heat transfer from memory, but will generally

check their own notes or a published source to find the most appropriate for a

particular application. Students should NOT attempt to learn the correlations given in

this section.

• Have a knowledge of fouling and fouling mechanisms

• Be able to calculate radiative heat transfer for simple geometries

2.1 One-Dimensional Conduction

The governing equation for one dimensional conduction is:

dxdtkAQ −=& (2.1)

which, for constant area, for example through a plane wall, may be integrated:

( 12 ttkAQ −−=& ) (2.2)

At this stage it is also convenient to introduce an equation relating heat transfer

between a fluid and a solid boundary:

tAQ ∆−= α& (2.3)

2.1

where α is the convective heat transfer coefficient. We will discuss the evaluation of

heat transfer coefficients in Chapter 4

In both cases the negative sign signifies heat transfer is from the higher temperature

to the lower temperature.

We shall look firstly at how these equations may be combined to give us a

relationship between heat transferred from one fluid stream to another, the

appropriate heat transfer coefficients, and the geometry of the barrier between the

fluids.

2.2 Heat Transfer between Two Fluids

In many applications two fluids are separated by a solid boundary, the two simplest

situations are shown in figures 2.1(a) and 2.1(b), representing fluids flowing on either

side of a flat plate and on the inside and outside of a tube, respectively.

a) Plane wall

b) Tube

Hot fluid inside

Hot fluid outside

Th

To

Ti

Two

Ti

Tc

Tw1Tw2

Two

Two

Twi

Twi

t

ri

ro

Figure 2.1 Heat Transfer through plate and tube

2.2

There are several resistances to heat transfer between the bulk of the hot fluid and

the bulk of the cool fluid as shown in Fig 2.1. For steady state conditions, the rate of

heat transfer from the hot fluid to the wall, the rate of heat flow through the wall,

and the rate of heat transfer from the wall to the cool fluid must be equal, defining

this as Q , the surface area of the wall as A, and the heat flux by & & &q Q A= ,we can

write:

( )&Q A T Ta h w= − −α 1 or ( )&q T Ta h w= − −α 1 (2.4a)

( )&QkAt

T Tw w= − −1 2 or (&qkt

T Tw w= − −1 2 )

T

(2.4b)

( )&Q A Tb w c= − −α 2 or ( )&q Tb w cT= − −α 2 (2.4c)

(The negative signs are included to indicate that energy flows from hot to cold)

Rearranging equations 2.4 gives

(&qT T

ah wα

= − − 1 )

(&qtk

T Tw w= − −1 2 ) (2.5)

(&qT T

bw cα

= − −2 )

Adding these equations gives:

( chba

TTqkt

−−=⎟⎟⎠

⎞⎜⎜⎝

⎛++ &αα11 )

)

)

(2.6)

which is equivalent to:

(&q U T Th c= − − (2.7)

or

(&Q UA T Th c= − − (2.8)

with

1 1 1U

tka b

= + +⎛⎝⎜

⎞⎠⎟

α α (2.9)

For a multi-layered wall this analysis may easily be extended to give:

2.3

1 1 11U

tka

i

ii

i n

b

= + +⎛⎝⎜

⎞⎠⎟

=

=

∑α α (2.10)

Examples of this situation encountered in practice, include be the wall of a building

comprising structural decorative and insulating layers and heat exchanger plates.

By analogy to the analysis of electrical circuits, we may consider the heat flux q as

analogous to the current I flowing through a series of resistances, r, and the driving

temperature

&

T∆ difference as equivalent to a voltage drop, V,. With reference to the

diagram below:

V2 V1 r1 r2 r3 rn

nrrrRR

VVI ............ where 2121 ++=

−=

The heat transfer through a wall of unit area may be expressed:

&qT T

Rh c= −−

(2.11)

RU

tk

tk

tk

r r r r ra

i

i

n

n b

a i

=

= + + + + + +

= + + + + +

1

1 11

1

1

α α...... .............

.......... .................. n b

(2.12)

Where the individual rs represent the heat transfer resistance for each boundary.

Often the values of these resistances differ by an order of magnitude or more and it

is permissible to neglect the smaller resistances.

In building design, for example, it is common to encounter structures having multiple

layers, some primarily structural or aesthetic, others providing thermal insulation.

2.4

There are also situations when intermediate temperatures are known, or must be

calculated. For example if the internal surface temperature of the wall illustrated in

Fig. 2.1(a) is known then equations 2.4(b) and (c) may be used to give:

( )cwb

TTqkt

−−=⎟⎟⎠

⎞⎜⎜⎝

⎛+ 1

1&

α (2.6(a))

Obviously it is undesirable to have a high thermal resistance in a heat exchanger, so

it would be unusual to have a composite wall separating the fluids by design. Fouling,

however, may lead to the deposition of a layer of material with poor thermal

conductivity on one or both sides of the wall. Hence, we must often consider the

situation where an additional fouling resistance must be included in equation 2.9.

( ch

bfb

t

wfa

a

TTr

kt

rq −

++++−=

α1

α1

1& )

)

(2.13)

The terms rfa and rfb representing the fouling resistances, or fouling factors on each

side of the wall.

Even for this simplest of geometries we can write this expression incorporating

fouling resistances in a number of ways, all of which are equally valid, for example:

(&Q U A T Tf h c= − − (2.14)

1 1U U

r rf

fa fb= + +⎛⎝⎜

⎞⎠⎟ (2.15)

or

1 1 1U

tkf af

w

t b= + +α fα

(2.16)

with the appropriate heat transfer coefficient, α f , for the fouled surface being

calculated from:

1 1α αf

fr= + (2.17)

2.5

To determine the heat transfer through a tubular element we follow a similar line of

reasoning. As for the plane wall, during steady state conditions the rate of heat flow

into a section of wall must equal the heat flow out at the other side. Furthermore, if

the tube is long and longitudinal temperature gradients are small, we can assume that

heat flow is one dimensional and in the radial direction.

Firstly consider the heat flow and temperature distribution within the tube wall:

For an element radius r and thickness δ r ,

&Q kATr

= −δδ

(2.18)

or, in the limit:

&Q kAd Td r

= − (2.18a)

For a length l of tube: A r= 2 lπ

Therefore

&Q k rld Td r

= − 2π (2.19)

This may be integrated between the limits ri and ro to give:

( )&

lnQ

kl T Trr

o i

o

i

= −−

⎛⎝⎜

⎞⎠⎟

2π (2.20)

Now if we consider the internal and external thermal resistances we can write a set

of equations:

(&Q lr T To o o wo= − −2π α ) (2.21a)

(&

lnQ

klrr

T To

i

o i= −⎛⎝⎜

⎞⎠⎟

−2 )π

(2.21b)

(&Q lr Ti i wi i= − −2π α )T (2.21c)

Which may be combined to give:

2.6

( )&

lnQ

r l klrr r l

T To o

o

i i io i2

1 1 1π α α

+⎛⎝⎜

⎞⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟ = − − (2.22)

As for the wall we may define an overall heat transfer coefficient or “U-value” such

that:

(&Q UA T To i= − − ) (2.23)

and

1 12

12

12UA r l kl

rr ro o

o

i i= +

⎛⎝⎜

⎞⎠⎟ +

π α π π αln

li

l

(2.24)

This implies that, with reference to the internal surface, having area A ri i= 2π ,

1 1U

rr

rk

rri

i

o o

i o

i i= + +

αln

α

l

(2.25)

and, with reference to the external surface, having area A ro o= 2π ,

1 1U

rk

rr

rro

i

o

o o

i

o

i i= + +α α

ln (2.26)

To avoid confusion regarding the area used , it is also possible to work in terms of

unit length:

( )&

lnQl r k

rr r

T To o

o

i i io i

12

12

12π α π π α

+⎛⎝⎜

⎞⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟ = − − (2.27)

leading to

(&Q U l T To i= − ′ − ) (2.28)

where

1 12

12

12′

= +⎛⎝⎜

⎞⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟

U r krr ro o

o

i iπ α π π αln

i (2.29)

Before considering how we might incorporate a fouling factor into a tubular

geometry it is worthwhile to look at the above expressions more closely, and, in

particular, examine what happens as ro approaches ri, i.e. the tube wall becomes thin

compared to the radius of the tube.

2.7

For a thin walled tube the mean radius may be calculated as ( )r r rm o i= + 2 , the area

is then given by A A A ri o m≈ ≈ = 2 lπ and the tube wall may be considered as a flat

wall, thickness t r ro i= − . The overall heat transfer coefficient and rate of heat

transfer may thus be calculated:

1 1 1U

tko i

= + +⎛⎝⎜

⎞⎠⎟

α α (2.30)

Which is identical to equation 2.6.

If the tube is thin walled and the thermal resistance, t/k, is small compared with the

two film resistances:

1 1 1U o i

= +⎛⎝⎜

⎞⎠⎟

α α (2.31)

and

(&Q U r l T Tm o i= − −2π ) (2.32)

The engineer must make a judgment as to whether these approximations are

reasonable for a given situation.

If fouling resistances are to be included, we have already seen that these may be

incorporated in the expression for the overall heat transfer coefficient through a

plane wall by simply adding the resistances. For the thin walled tube this may be

expressed:

1 1 1U

rtk

rf o

foi

i= + + + +⎛⎝⎜

⎞⎠⎟

α α (2.33)

or:

1 1 1U

tkf o

=′+ +

′⎛⎝⎜

⎞⎠⎟

α iα (2.34)

where

1 1′= +

α αrf (2.35)

2.8

It is convenient to use this modified heat transfer coefficient when dealing with thick

walled tubes, and as we shall see later, with all geometries where the heat transfer

areas differ for each stream.

For the thick walled tube we can, for example, write:

1 1′=

′+ +

′Urk

rr

rro

i

o

o o

i

o

i iαln

α (2.36)

EXTENDED SURFACES - FINS

We have seen that the rate of heat transfer from a plane surface is proportional to

the surface area, an obvious technique for increasing heat transfer is to increase the

surface area available. This may involve using more or longer tubes in a heat

exchanger or by adding to the surface area using fins. In applications where the

geometry is fixed – for example the top of a microprocessor or the cylinder of an

air-cooled engine, the use of an extended surface is the only option.

Figure 2.2 Typical Fin Types

he surface area on one or both sides of a heat exchanger may be increased by the

use of extended surfaces or fins. There is a wide range of geometries employed in

extending the surface in contact with a fluid. Surfaces which are separated from the

2.9

other fluid only by the thin layer of material through which conduction occurs (e.g.

the plane wall or tube discussed above) are referred to as the primary surface of a

heat exchanger. Additional surface which is in contact with one fluid but from which

there is a tortuous conduction path to the other fluid is known as the secondary

surface.

We will first analyse the simplest fin type and then discuss how the results may be

used in more complex geometries. We shall consider a rectangular fin on a plain

surface as shown in Figure 2.3.

Figure 2.3 Diagram of heat flow in rectangular fin

If the length, l, and breadth, L, of the fin are large compared to the thickness, b, we

can assume that conduction through the fin is approximately one dimensional.

The heat flow into the element dx at some position, x, from the root of the fin is

given by:

&Q kAdTdx

kLbdTdxx = −

⎛⎝⎜

⎞⎠⎟ = −

⎛⎝⎜

⎞⎠⎟ (2.37)

and the heat flow out of the element is:

2.10

( )& &

&Q Q

dQdx

dx kLbdTdx

ddx

kLbdTdx

dx

kLbdTdx

kLbd Tdx

dx

x dx x+ = +⎛⎝⎜

⎞⎠⎟ = −

⎛⎝⎜

⎞⎠⎟ −

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

= −⎛⎝⎜

⎞⎠⎟ −

⎛⎝⎜

⎞⎠⎟

2

2

(2.38)

The difference between the inflow and outflow by conduction must be equal to the

net outflow of heat from the element to the surroundings:

( )dQ Q Q kLbd Tdx

dxx x dx& & &= − =

⎛⎝⎜

⎞⎠⎟+

2

2 (2.39)

and, for surroundings at Ts this is also given by:

( )( ) bLTTLdxATTdxbLATTAQd sss >>−≈−+=−= since )(2)(2)( ααα& (2.40)

defining so that (θ = −T Ts )d Tdx

ddx

2

2

2

2=θ

, since Ts is constant, and equating the two

expressions for heat loss from the element:

ddx kb

2

22θ α

θ= (2.41)

This differential equation has a general solution of the form:

θ = + −Me Nemx mx (2.43)

where

mkb

=2α

The values of the constants M and N are then determined with reference to

appropriate boundary conditions:

At the root of the fin x=0 and the fin temperature is equal to the root temperature,

To.

θ0 0= − = +T T M Ns (2.44)

The second boundary condition is less obvious, but if the fin is very slender so that

the heat loss from the tip can be neglected, or if the fin is insulated, then, at x=l,

2.11

Q kLbddx

ddxl

l l

= −⎛⎝⎜

⎞⎠⎟ =

⎛⎝⎜

⎞⎠⎟ =

θ θ0 0, or (2.45)

Differentiating equation 2.43 and putting x=l gives:

Mme Nmeml ml− =− 0 (2.46)

Combining equations 2.44 and 2.46 gives values for the constants M and N:

Me

e eN

ee e

ml

ml ml

ml

ml ml=+

=+

−

−θ θ0 0 and − (2.47)

Substituting these values in equation 2.43 gives:

( ) ( )( )( )θ θ θ=

++

⎛

⎝⎜

⎞

⎠⎟ =

−⎛⎝⎜

⎞⎠⎟

− − −

−0 0

e ee e

m l xml

m l x m l x

ml ml

( ) coshcosh

(2.48)

The heat flow from the surface of the fin is equal to the heat flow through the base

of the fin, therefore at xo, using equation 2.45, gives:

( )( )( )

( )

& sinhcosh

tanh

Q kLbdtdx

mkLbm l x

ml

mkLb mlx

00

00

0

= −⎛⎝⎜

⎞⎠⎟ =

−⎛

⎝⎜

⎞

⎠⎟

==

θ

θ

(2.49)

Now an ideal fin would have infinite thermal conductivity, hence the entire fin would

have a surface temperature equal to the temperature of the root, and the rate of

heat transfer from the fin would be given by:

&Q A Lideal = =α θ α θ0 2 l 0 (2.50)

Defining the fin efficiency as:

ηfin =Rate of heat transfer from fin

Rate of heat transfer from ideal fin of the same geometry (2.51)

gives:

2.12

( )η

θα θfin

mkLb mlLl

= 0

02tanh

(2.52)

Remembering that mkb

=2α

, we can write equation 2.52 as:

( )ηfin

mlml

=tanh

(2.53)

If fouling occurs the fouling resistance should be taken into account when evaluating

the heat transfer coefficient used in determining m. i.e.

1 1′= +

α αrf (2.54)

It is implicit in the above analysis that the tip of the fin is adiabatic. This

approximation holds if the tip of the fin is not insulated or if it butts on to the tip of

an adjacent fin. However, if there is heat transfer from the tip of the fin then this

may be taken into account by correcting the length of the fin by adding 1/2b, i.e.

lc=l+1/2b. The corrected length may then be used in both the evaluation of fin

efficiency and fin area.

Expressions, often presented in graphical form, are available for the fin efficiency of

many shapes of fin. Examples are given in Figures 2.4 and 2.5.

In order to use an expression of the form:

&Q UA T= ∆

to determine the rate of heat transfer across a boundary which includes extended

surfaces we can derive an appropriate equation, following the example of equations

2.4 and 2.21;

( ) ( )

( ) ( )owooo,fino

w

wowi

wiiic,fini

TTmQ

rTTQ

TTmQ

−′−=

−−=

−′−=

area unfinned + area fin x

area unfinned + area fin x

α

α

&

&

&&

(2.55)

2.13

which may be rearranged to give:

( )( ) ( )

ofinow

ifini

oi 1r1Q

TTarea unfinned + area fin x area unfinned + area fin x ηαηα ′

++′

=−

−&

(2.56)

or

( ) ( )( )oi

ofinow

ifini

TT1r1

1Q −

′++

′

=

area unfinned + area fin x area unfinned + area fin x ηαηα

&

(2.57)

so

( ) ( )1

ofinow

ifini

1r1UA−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

′++

′≡

area unfinned + area fin x area unfinned + area fin x ηαηα (2.58)

As we shall see later, the product UA is an important parameter in heat exchanger

design.

In applications where fins are employed, it is likely that the purpose is to increase

heat transfer (or reduce temperature difference) It is therefore desirable to make

each term on the right hand side of equation 2.58 small. Normally rw is the smallest

of the terms and, for calculation purposes, may often be neglected. It is more

effective to reduce the larger of the two remaining resistances, especially if they

differ significantly. Therefore, if finning is to be applied to one surface only, it should

be applied to the side with the lower heat transfer coefficient (unless fouling,

corrosion or other considerations render this impracticable). This is clearly seen in

liquid to gas heat exchangers where it is common for the liquid (or evaporating or

condensing fluid) flows inside tubes which are externally finned.

2.14

Figure 2.4 Efficiency of Radial Fins - constant cross section (SI units with l,b in meters)

Figure 2.5 Efficiency of Axial Fins (SI units with l,b meters)

2.15

2.3 Convective heat transfer

We have already applied the relationship:

( )&Q A t A T Tfluid wall= − ≡ − −α α∆ (2.3)

to determine the rate of heat transfer from a surface of area A and at temperature

Twall to a fluid at temperature Tfluid. Representative values of the heat transfer

coefficient, α, may be used in preliminary designs, but it is important that we should

be able to estimate values of α for a particular geometry and flow conditions with

some accuracy if we are to have confidence in a heat transfer calculation, whether it

be a heat exchanger design or calculation of a component temperature. There are

few conditions for which analytical solutions for α are available, and these are not

often encountered in engineering situations. It is therefore usually necessary to rely

upon empirical correlations in the determination of convective heat transfer

coefficients.

Convection may be forced or free, free convection is also known as natural

convection. In forced convection an external driving force (e.g. a pump or fan) causes

he movement of fluid over the heat transfer surface. In free or natural convection

the fluid movement is induced by the heat transfer and the resulting density change

within the fluid. If the fluid movement induced by density changes is significant

compared to forced fluid movement heat transfer is said to be by mixed convection.

The majority of heat exchangers operate with forced convection. The exceptions

being most condensers and some boilers We shall look at single-phase forced

convection, boiling and condensation in some detail. While the importance of natural

convection in many applications (e.g. electronics cooling and space heating) should

not be underestimated it is rarely encountered in heat exchangers.

Empirical correlations are based upon experimental observations. While the form of

the correlations may have some theoretical or conceptual justification, their accuracy

relies upon the reliability of the experimental observations and the similarity of the

experimental conditions and those to which the correlation is to be applied. It is

obvious that correlations are geometry dependent (e.g. applying to flow inside tubes,

crossflow outside tubes or over flat plates) There are other, less obvious

2.16

transitions, the most important being between laminar and turbulent flow and the

occurrence or otherwise of a phase change. From the designer’s point of view it is

essential that an appropriate correlation is applied, use of a correlation

outside the range of conditions for which it has been experimentally verified is very

dangerous.

2.17

2.3.1 Dimensionless groups and units

Quantity Symbol(s) Dimensions SI Units

Length L,x L m

Time t t s

Mass M M kg

Temperature T,t,θ T K

Absolute Viscosity µ ML-1t-1 kg/ms

Acceleration a Lt-2 m/s2

Coefficient of Expansion β T-1 K-1

Density ρ ML-3 kg/m3

Enthalpy H ML2t-2 J

Force F MLt-2 N

Heat Q ML2t-2 J

Heat Flux &q Mt-2 W/m2

Heat Transfer Coefficient h,α Mt-3T W/m2K

Internal Energy U ML2t-2 J

Kinematic Viscosity ν=µ/ρ L2t-1 m2/s

Mass Flow Rate &m Mt-1 kg/s

Mass Flux G Mt-1L-2 kg/m2s

Power &W ML2t-3 W

Pressure p Mt-2L-1 N/m2

Shear Stress τ Mt-2L-1 N/m2

Specific Enthalpy h L2t-2 J/kg

Specific Heat Capacity c,cp,,cv L2t-2T-1 J/kgK

Surface Tension σ Mt-2 N/m

Thermal Conductivity k.λ MLt-3T-1 W/mK

Thermal Diffusivity a,α L2t-1 m2/s

Thermal Resistance r,R Tt3M-1 m2K/W

Velocity V,U,v Lt-1 m/s

Volume V L3 m3

Work W ML2t-2 J

Table 2.1 Quantities, units and dimensions

Table 2.1 lists various quantities and their dimensions commonly encountered in heat

transfer calculations. Heat transfer data and correlations are frequently presented in

the form of dimensionless groups. Some of the dimensionless groups which we will

deal with are listed below. Other groups will be introduced when we consider two-

2.18

phase heat transfer. The student should verify that the groups below are indeed

dimensionless. It is possible to devise other groupings by combining the common

groups.

Dimensionless Groups

Darcy friction factor fV

cof= =

842

τρ

(2.59(a))

Fanning skin friction coefficient cV

ff

o= =τρ1

22 4

(2.59(b))

Colburn j factor jh = St Pr23 (2.59(c))

Grashof Number Gr =g TLβν∆ 3

2 (2.59(d))

Nusselt Number Nu =αdk

e

fluid (2.59(e))

Prandtl Number Pr =µck

p

fluid (2.59(f))

Rayleigh Number Ra = GrPr (2.59(g))

Reynolds Number Re = = =ρµ νVd Vd Gde e

µe (2.59(h))

Stanton Number Stq

Vc T Vcp p= = =

&

ρα

ρ∆Nu

RePr (2.59(i))

In evaluating the above groups there is often some ambiguity in the choice of values

which must be resolved. The physical dimension for flow over plates is generally

taken as the distance along the plate, for flow in or around ducts it is the hydraulic

or equivalent diameter defined:

de =4 x Cross sectional area

Wetted Perimeter (2.60)

As expected, for a circular duct or pipe, diameter d, this is given by:

ddd

de =4 42ππ

= (2.61(a))

For a square duct, side length x,

2.19

dxx

xe = =44

2

(2.61(b))

and for a rectangular duct, width a and depth b:

( )dab

a be = +4

2 (2.61(c))

if , for example closely spaced plates, this becomes: a >> b

( ) b2a2ab4

ba2ab4de =≈+

= (2.61(d))

i.e. the equivalent diameter is equal to twice the plate spacing.

For flow through an annulus having inner and outer diameters d1 and d2, respectively

the hydraulic diameter may be calculated:

( )( )

( )( )( ) (d

d dd d

d d d dd d

d de =−

+=

− +

+= −

4 4 4 422

12

2 1

2 1 2 1

2 12 1

ππ

ππ

) (2.61(d))

which is equal to twice the thickness of the annular gap.

The hydraulic radius, re,, is defined:

rd

ee=

Cross sectional areaWetted Perimeter 4

= (2.62)

Thermophysical fluid properties (density, viscosity etc.) vary with temperature. It is

important, particularly if the temperature difference between the wall and the fluid is

large, that the appropriate temperature is chosen. Normally this is the fluid bulk

temperature, but some correlations require properties to be evaluated at the mean

film temperature:

TT T

filmfluid wall

=+

2 (2.63)

2.3.2 Single-phase convection.

As stated above convection may be free or forced. While free convenction is

important in many applications, few heat exchangers rely on single-phase free

2.20

convection. We will therefore concentrate on forced convection, however, for

completeness some correlations which may be used to determine free convection

heat transfer coefficients are briefly discussed. Whether the flow is induced by

natural convection or forced it can be described as either laminar or turbulent.

Laminar and turbulent flow

If one imagines a deck of playing cards or a sheaf of paper, initially stacked to

produce a rectangle, to be sheared as shown in Fig. 2.6, it can be seen that the

individual cards, or lamina, slide over each other. There is no movement of material

perpendicular to the shear direction.

Figure 2.6 Shear applied to parallel sheets

Similarly, in laminar fluid flow there is no mixing of the fluid and the fluid can be

regarded as a series of layers sliding past each other. If the flow is laminar a thin

filament of dye inserted in the fluid will remain as a thin filament as it follows the

flow.

Consideration of a simple laminar flow allows us to define viscosity. Fig. 2.7

illustrates the velocity profile for a laminar flow of a fluid over a flat plate:

2.21

v

y

Free stream velocity

Plate

Figure 2.7 Velocity profile in laminar flow over a flat plate

The absolute or dynamic viscosity of a fluid, µ, is defined by:

τ µ=dvdy

(2.64)

where τ is the shear stress. At the wall, the velocity of the fluid must be zero, and

the wall shear stress is given by:

τ µww

dvdy

=⎛⎝⎜

⎞⎠⎟ (2.65)

The kinematic viscosity of a fluid is defined:

νµρ

= (2.66)

(Be careful not to confuse ν and v!! )

In practice, laminar flow is observed at low speeds, in small tubes or channels, with

highly viscous fluids and very close to solid walls.

If the fluid layers seen in laminar flow break up and fluid mixes between the layers

then the flow is said to be turbulent. The turbulent mixing of fluid perpendicular to

the flow direction leads to a more effective transfer of momentum and internal

energy between the wall and the bulk of the fluid. Turbulent flow is the more

common regime for bulk flow in most heat transfer equipment, but laminar flow is

2.22

encountered in highly compact heat exchangers and those handling very viscous

fluids. Even when the bulk of the flow is turbulent a very thin laminar layer exists

close to the wall, this is important when considering processes close to the wall.

It should come as no surprise to the student that the heat transfer characteristics of

laminar and turbulent flows are very different. In forced convection the magnitude of

the Reynolds number provides an indication of whether the flow is likely to be

laminar or turbulent:

For flow over a flat plate, as shown in Fig 2.8 we may determine whether the flow in

the boundary layer is likely to be laminar or turbulent by applying the following

conditions:

Re x

V x=⎛⎝⎜

⎞⎠⎟ <∞ρ

µ105 Laminar flow

(2.67)

Re x

V x=⎛⎝⎜

⎞⎠⎟ >∞ρ

µ106 Turbulent flow

where x is the distance from the leading edge of the plate.

Laminarsublayer

Laminar Transition Turbulent

x

V∞

Figure 2.8 Development of the boundary layer over a flat plate

For values of Reynolds number between 105 and 106 the situation is complicated by

two factors. Firstly, the transition is not sharp, it occurs over a finite length of plate.

In the transition region the flow may intermittently take on turbulent and laminar

characteristics. Secondly, the position of the transition zone depends not only upon

the Reynolds number, it is also influenced by the nature of the flow in the free

2.23

stream and the nature of the surface. Surface roughness or protuberances on the

surface tend to trip the boundary layer from laminar to turbulent.

For flow in pipes, channels or ducts the situation is similar to that for a flat plate in

the entry region, but in long channels the boundary layers from all walls meet and

fully developed temperature and velocity profiles are established.

For fully developed flow in pipes or channels the transition from laminar to turbulent

flow occurs at a Reynolds number, based on the channel hydraulic diameter of

approximately 2000. As with the boundary layer on a flat plate, the transition may

occur at higher or lower values of Red. If the flow at entry to the channel contains no

turbulence and the channel is very smooth, laminar flow may be sustained up to

Reynolds numbers of 5-10000. Turbulence may occur at values of Red as low as 1000

but at low Reynolds numbers may decay if induced by, for example, sharp corners.

As we shall see, heat transfer coefficients are generally higher in turbulent flow than

in laminar flow, and higher in the entry region than in the fully developed region.

Heat exchanger designers may therefore incorporate features which either promote

turbulence or lead to a geometry which approximates to many short channels. The

velocity distribution and variation in local heat transfer coefficient observed at entry

to a tube at Red>>2000 is illustrated in Fig. 2.9.

2.24

Figure 2.9 Velocity distribution and variation of local heat transfer coefficient for turbulent flow near the entrance of a uniformly heated tube

Laminar forced convection in ducts1

Examination of the velocity and thermal boundary layers permits the development of

analytic solutions for heat transfer between a wall and a fluid in laminar flow, at least

in simple geometries. It is beyond the scope of these notes to derive the solutions,

however some of the more useful are presented in Table 2.2.

Three values of Nusselt number are given for each geometry, the appropriate value

depending upon the boundary conditions. These are:

• NuH1 =Average Nusselt no. for uniform heat flux in flow direction and

uniform wall temperature around perimeter at any cross section

• NuH 2 =Average Nusselt no. for uniform heat flux both axially and

around the perimeter

• NuT = Average Nusselt no. for uniform wall temperature.

also tabulated are values of the product fRed.

1 The term ducts here encompasses tubes and channels

2.25

Table 2.2 Nusselt number and friction factor for fully developed flow in ducts

2.26

Table 2.3 gives values of the Nusselt number for heat transfer to or from laminar

flow in an annulus with one wall insulated (adiabatic) and the other is maintained at

constant temperature.

Table 2.3 Nusselt number and friction factor for fully developed flow in annuli

Particularly in compact heat exchangers and heat sinks on microelectronic systems,

the effective duct length may be quite short and entry effects must be taken into

consideration. Analytic and empirical solutions for the variation of heat transfer in

the entry region are available. Typically these are presented as either:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

PrRef

NuuN

developedfully dh

hdx

or

⎟⎟⎠

⎞⎜⎜⎝

⎛=

PrRef

NuNu

developedfully

x

dh

hdx

where Nu is the mean Nusselt number from duct entry to a position x along the

duct and Nux is the local Nusselt number at a distance x from the entry.

Nufully developed is the Nusselt number for fully developed flow for the corresponding

boundary conditions. An example of entrance length effect is given in 2.10.

2.27

Figure 2.10 Ratio of mean Nusselt number from entry to x to fully developed Nusselt number ( constant temperature wall)

The thermal entry lengths, Le, for the simultaneously developing hydroynamic and

thermal profiles in laminar flow are given by:

LdLd

e

hdh

e

hdh

≈

≈

0 037

0 053

. Re Pr

. Re Pr

(Uniform surface temperature)

(Uniform heat flux) (2.67)

Turbulent forced convection in ducts

As we have already seen, the Reynolds Number is a particularly important group

when dealing with forced convection. The value of the Reynolds Number may be

used to determine whether the flow is laminar or turbulent. The Reynolds number is

also included in most turbulent flow heat transfer correlations, many of which are

expressed in the form:

Nu=f(Re,Pr, fluid property correction)

For example, the heat transfer coefficient in single-phase turbulent flow is commonly

determined from the Dittus-Boelter equation:

2.28

Nu = 0 023 0 8. Re Pr. n (2.68)

n Tn T

w f

w f

= >

= <

0 40 3..

for heating for cooling

TT

with properties evaluated at the bulk or mean bulk fluid temperature. This gives

results for Nu within 20% for uniform wall temperature and uniform wall thickness

conditions within the following ranges.

05 1206000 1060

7

. PrRe/

< <

< <<

d

eL D

A modification to equation 2.68, taking into account the change in viscosity with

temperature in the thermal boundary layer is:

Nu =⎛

⎝⎜

⎞

⎠⎟0 027 0 8

0 14

. Re Pr.

.

n

s

µµ

(2.69)

If evaluating local Nusselt number then the bulk fluid temperature is equal to the

local bulk temperature, i.e. the temperature which would be measured if the fluid at

that station were to be fully mixed. If evaluating the mean Nusselt number over a

length of tube the mean bulk temperature is given by:

TT T

mm in m out=

+, ,

2 (2.70)

Reynolds Analogy

Let us consider a turbulent flow past a wall as shown in Fig. 2.9

2.29

Free stream at V, Tf

Wall at Tw

m

Assume that in unit time, over an area A, a mass m, of fluid moves from the free

stream to the wall and a corresponding mass moves away from the wall. Assuming

that the fluid at the wall is stationary and reaches thermal equilibrium with the wall

we can say:

Transfer of momentum from the fluid to the wall in flow direction = ( )m V − 0

Transfer of heat from the wall to the fluid = ( )− −mc T Tp f w

Remembering that

force = rate of change of momentum

and the shear stress on the wall is equal to the force exerted by the fluid in the flow

direction per unit area:

( )Rate of heat transfer /Rate of momentum transfer /

AA

q c T TVw

p f w= = −

−&

τ (2.71)

Rearranging and putting ( )∆T T Tw f= −

ατ

= =&qT

cVp w

∆ (2.72)

showing that the heat transfer coefficient and wall shear stress are closely related. In

dimensionless form we can write equation 2.72 as:

αρ

τρVc Vp

w=12 1

22 (2.72)

Using the groups defined in equation 2.59:

St =Nu

RePr=

12

cf (2.73(a))

or

Nu =12

RePrcf (2.73(b))

2.30

We will discuss the pressure drop through ducts pipes and fittings in Section 4, but

at this stage it is worth noting that a force balance on a length, L, of circular pipe as

shown in Fig. 2.12 gives:

( )p p A Ac w1 2− = τ s (2.74)

p1Vm

p2VmτwAc

L

Figure 2.12. Nomenclature used in equation 3.74

Substituting expressions for the cross sectional area, Ac, the surface area, As, and the

wall shear stress from equation 2.59(b)

∆

∆

pD

V c DLρ π

pV c LD

m f

m f

π

ρ

22

24

12

2

=

=

(2.75)

Thus, if the heat transfer coefficient and wall shear stress are related, the pressure

drop and heat transfer coefficient are also closely linked.

For turbulent flow in smooth pipes with Reynolds Number up to 105 the Blasius

equation, equation 2.76, gives reasonable results for friction factor, cf, gives:

cf = 0 079 0 25. Re . (2.76)

Reynolds analogy (as presented in equation 2.73(b)) thus suggests:

Nu = 0 0395 0 75. Re Pr. (2.77)

which, for Pr=1 gives very similar results to the Dittus Boelter equation. This is

illustrated in Fig.2.11 which shows calculated values of Nusselt number using Dittus

Boelter equation and the combination of Reynolds analogy and the Blasius equation

as presented in equation 2.77. The Nusselt number for laminar flow with constant

heat flux is also included on Fig. 2.13. It is clear that, for a given Reynolds number the

2.31

value of estimated Nusselt number differs significantly depending whether the flow is

assumed to be turbulent or laminar. This confirms the importance of ensuring that

the appropriate flow type is identified. It also shows that heat transfer enhancement

may be achieved for Reynolds numbers from c1000 to the transition region by

“tripping” the flow to induce turbulence.

The Prandtl Number is the ratio of molecular momentum diffusivity to the thermal

diffusivity of a fluid. It is therefore to be expected that Reynolds Analogy is only valid

for Pr ≈1, since the derivation implied equal momentum and thermal diffusivity.

Alternative correlations

While the Dittus Boelter correlation is widely used, its accuracy is limited. A more

complex (and thus more awkward to use!) correlation is that due to Gnielinsky, this

is regarded as having an accuracy within 6%. With all properties evaluated at the

mean bulk temperature:

For 0<dh/L<1, 0.6<Pr<2000, Redh>2300

( )( )( ) ( )

( )

Nudhdh

dh

=−

+ −+⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= −−

f

f

dL

f

h8 1000

1 12 7 8 11

0 79 164

12

23

23

2

Re Pr

. Pr

. ln Re .

(2.78)

approximations to equation 2.78 may be used over the appropriate ranges:

For 0.5<Pr<1.5, 2300<Redh<106, 0<dh/L<1

( )Nu dh dh0.8= − +

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0 0214 100 10 4

23

. Re Pr . dL

h (2.79(b))

For 1.5<Pr<500, 2300<Redh<106, 0<dh/L<1

( )Nudh dh0.87= − +

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0 012 280 10 4

23

. Re Pr . dL

h (2.79(c))

Equations 2.78 and 2.79 give mean Nusselt numbers over the length of the tube. If

applied to fully developed conditions local Nusselt numbers may be obtained by

2.32

setting dL

h to zero. Gnielinsky recommended that a further correction may be

included to take into account property variations due to temperature. The Nusselt

no calculated using equation 2.78 or 2.79 above should be multiplied by:

TT

b

s

b

s

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

0 45 0 11. .PrPr

for gases or for liquids.

The Gnielinsky correlation was derived for uniform wall temperature, but gives good

results for uniform heat flux conditions. An alternative correlation, based on data for

uniform heat flux was proposed by Pethukov for fully developed flow:

( )( ) ( )

( )

Nudhdh

dh

=+ −

⎛⎝⎜

⎞⎠⎟

= −−

f

f

f

s

n8

107 12 7 8 1

0 79 164

12 2

3

2

Re Pr

. . Pr

. ln Re .

µµ (2.80)

Pethukov’s correlation applies when all properties (except µs ) are evaluated at the

bulk temperature and values of the constants used are:

n = 0.11 (liquids, heating)

n = 0.25 (liquids, cooling)

n = 0 (gases)

104<Redh<5 x 106, 0.8< (µ µs ) <40

0.5<Pr<200 6% uncertainty

200<Pr<2000 10% uncertainty

The Dittus Boelter, Gnielinsky and Pethukov correlations apply to a wide range of

fluids, but are inappropriate for liquid metals (Pr<0.1).

2.33

Forced convection over cylinders, rods and tube banks

The boundary layer observed when a fluid flows over a cylinder or rod cannot be

uniquely described as laminar or turbulent. The boundary layer itself is laminar at the

front of the cylinder and, depending upon the Reynolds number, may become

turbulent. Additionally, at all but the lowest Reynolds numbers, the boundary layer

separates from the surface of the cylinder at some point and a wake is formed. The

wake may be laminar or turbulent.

Data for air (Pr=approximately 0.7) has been correlated in a form:

Nu cdmd= Re Prn (2.81)

The value of n in equation 2.81 is 0.33. Properties are evaluated at the mean film

temperature Values of c and m are tabulated in Table 2.4

Red c m

4-35 0.895 0.384

35-5000 0.657 0.471

5000-50,000 0.167 0.633

50,000-230,000 0.0234 0.814

Table 2.4 Values of c and m for use in equation 2.81

It is recommended that the use of equation 2.81 and the constants of Table 2.4 be

restricted to the range 0.5<Pr<10.

For long bars of non cross circular cross section, equation 2.81 may be used with the

characteristic dimension and constants listed in Table 2.5

A correlation based on a wider range of data and valid for RedPr>0.2 and with

properties evaluated at the mean film temperature is:

Red>400.000

2.34

( )Nud = +

+⎡⎣⎢

⎤⎦⎥

+⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0 30 62

1 0 41

282000

12

13

23

14

58

45

.. Re Pr

. Pr

Red d (2.82(a))

20,000<Red<400,000

( )Nud = +

+⎡⎣⎢

⎤⎦⎥

+⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

0 30 62

1 0 41

282000

12

13

23

14

12

.. Re Pr

. Pr

Red d (2.82(b))

Red<20,000

( )Nud = +

+⎡⎣⎢

⎤⎦⎥

0 30 62

1 0 4

12

13

23

14

.. Re Pr

. Pr

d (2.82(c))

Heat exchangers rarely comprise single tubes in crossflow, they usually incorporate

tube banks and the flow differs from that around a single tube in two ways:

• The velocity between the tubes is greater than the free stream velocity

• The flow field on a tube row is influenced by the presence of upstream row(s)

Tube banks may be in-line or staggered, as shown in Fig.2.14 (a) and (b) respectively.

Zukauskas recommends that a correlation of the form:

Nu cdm n

sd=

⎛

⎝⎜

⎞

⎠⎟Re Pr

PrPr

.0 25

(2.83)

should be applied to determine the mean Nusselt no (and hence heat transfer

coefficient) tube banks having more than 16 rows. The velocity used in evaluating the

Reynolds number is the maximum fluid velocity in the bank. All properties are

evaluated at the mean bulk temperature, with the exception of Prs which is

evaluated at the surface temperature of the tubes.

Values of c, m and n for use in equation 2.83 are given in Table 2.6.

2.35

V∞,T∞

d

ST

SL

Column

Row

V∞,T∞

d

SD

SL

Column

Row

XY

ST

(b) staggered

(a) in line

Figure 2.14 Tube bank arrangements

2.36

Red c m n

10-100 0.9 0.4 0.36

100-1000 0.52 0.5 0.36

1000-200,000 0.27 0.63 0.36

200,000-2,000,000 0.033 0.8 0.4

(a) in line arrangement

Red c m n

10-500 1.04 0.4 0.36

500-1000 0.71 0.5 0.36

1000-200,000 0.35 0.63 0.36

200,000-2,000,000 0.031 0.8 0.36

(b)staggered arrangement

Table 2.6 Constants for use in equation 2.83

For in line tube banks the maximum velocity may be calculated by considering

conservation of mass, assuming incompressible flow:

VV SS d

T

Tmax = −

∞ (2.84a)

For staggered tube banks the maximum velocity may occur either between adjacent

tubes in a row or between one tube and a neighbouring tube in the succeeding row,

i.e. through the planes X or Y marked on fig. (b). Conservation of mass, assuming

incompressible flow, gives:

( ) (V S V S d V S dT X T Y D∞ = − = −2 ) (2.84b)

and the maximum velocity is the larger of VX and VY.

Equation 2.83 is valid for 16 or more tube rows. For N rows, where N is less than

16, the mean Nusselt number should be reduced by a factor c1.

NuNu

cN

161= (2.85)

2.37

where c1 is given in Fig . 2.15.

Figure 2.15 Correction factor c1 for use in equation 3.85

Complex Geometries

For complex geometries it is unlikely that an appropriate correlation is available.

Experimental data for a number of configurations typical of those used in compact

heat exchangers has been published by Kays and London2. If new geometries are to

be developed it is likely that experimental measurements will be required to produce

a correlation. Manufacturers may publish such data, or they may be proprietary.

Sample figures from Kays and London are given as Figs 2.16(a)-(f) - unfortunately,

the data are in American units.

2 Kays W.M.and London A.L.,Compact heat exchangers, McGraw-Hill, 2nd Edition, 1964

2.38

Figure 2.16 (a) Heat transfer and friction factor for Plain plate-fin surface 9.03 (h=heat transfer coefficient)

f/4

f/4

Figure 2.16 (b) Heat transfer and friction factor for Plain plate-fin surface 11.1 (h=heat transfer coefficient)

2.39

f/4

Figure 2.16 (d) Heat transfer and friction factor for Plain plate-fin surface 6.2 (h=heat transfer coefficient)

f/4

Figure 2.16 (c) Heat transfer and friction factor for Plain plate-fin surface 5.3 (h=heat transfer coefficient)

2.40

f/4

Figure 2.16 (e) Heat transfer and friction factor for finned circular tubes, surface CF-7.34 (h=heat transfer coefficient)

2.41

2.4 Boiling and Evaporation

2.4.1 Introduction.

Many heat transfer applications involve the evaporation of a liquid. Boiling of a single

substance is a vital part of vapour power and refrigeration cycles. If we are to design

boilers or evaporators we must be able to determine the relationship between the

rate of boiling heat transfer, operating conditions and wall temperature for our heat

exchanger. In this introductory study. This introductory note is limited to the case of

boiling single fluids, but it should be remembered that evaporation occurs frequently

as part of a separation process, in which case the vapour formed has a different

composition from the boiling liquid

While the terms "boiling" and "evaporation" are used loosely to describe the action

of converting a liquid to a vapour by the transfer of energy to the liquid at its

saturation temperature it is necessary to be more precise when describing the

mechanisms involved. Boiling is the addition of heat causing liquid to evaporate and

the vapour to flow away from the heated surface. Evaporation is the conversion of

liquid to vapour which occurs at the liquid vapour interface.

Boiling is categorised according to the geometric situation and according to the

mechanism occurring. The geometric situations commonly encountered are:

Pool Boiling - this is defined as boiling from a heated surface submerged in a

stagnant pool of liquid. The only movement of the liquid being that induced by the

boiling process.

Flow Boiling - This is defined as boiling of a liquid as it is pumped through a heated

channel.

These are analogous to free and forced convection. Boiling outside tube bundles, for

example in a fire-tube boiler, combines elements of both situations- a recirculating

flow is induced through the bundle due to the vapour generation.

The three mechanisms of boiling which are observed are:

2.42

Nucleate Boiling- This involves the formation and growth of bubbles, usually on the

heated surface, the bubbles then leave the heated surface and rise to the surface of

the liquid. Fig.1 illustrates nucleate and film boiling.

Convective Boiling - This mechanism, sometimes referred to as evaporation,

involves transfer of heat from the heated surface through a thin layer of liquid and

evaporation of liquid at the liquid vapour interface.

Film Boiling- This mechanism occurs when the heated surface is blanketed by a film

of vapour, heat transfer is then by conduction through the vapour layer and

evaporation occurs from the liquid in contact with this liquid film.

Fig 2.17 illustrates nucleate and film boiling.

Nucleate Boiling Film Boiling

Figure 2.17 Schematic representation of film and nucleate boiling

The mechanisms involved in boiling are complex and the relationships used in

design and analysis are almost all empirical or semi-empirical, however, in

formulating and using empirical correlations it is necessary to have an understanding

of the underlying processes.

2.4.2 Pool Boiling

In 1934 Nukiyama performed a pool boiling experiment, passing an electric current

through a platinum wire immersed in water. The apparatus is shown schematically in

Fig. 2.18. The heat flux was controlled by the current through and voltage across the

wire and the temperature of the wire was determined from its resistance. Nukiyama

then proposed a boiling curve of the form shown in Fig. 2.19

2.43

Condenser

Heated Cylinder(or flat surface)

Vapour

Liquid

Figure 2.18 - Simple Pool Boiling Experiment

Since we have a liquid and vapour coexisting in the cylinder both must be at (or

during boiling, very close to,) the saturation temperature of the fluid at the pressure

in the container. If we measure the surface temperature of the heater, T, the

temperature of the fluid, Tsat, the rate of energy supply to the heater, and the

heater surface area, A, we may carry out a series of tests and plot a graph of

or more usually

& ,Q

log & ,Q

( )log & log &q Q= A against log∆Tsat , where , often

referred to as the wall superheat.

(∆T T Tsat sat= − )

As the heat flux, q , is increased while keeping the temperature of the fluid constant,

we would expect the temperature of the rod to increase. The designer of heat

transfer apparatus must be able to determine the relationship between heat flux and

temperature difference.

&

The relationship between heat flux and wall superheat for a typical fluid is shown

schematically in fig. 2.19.

2.44

( )log &q

( )log ∆Tsat

A B*B

C

DE

F

GG*

H

Figure 2.19 Schematic representation of boiling curve

For the case of controlled heat flux (for example, electric heating) the various

regimes may be described:

For increasing heat flux, in the region 'A'-'B' heat transfer from the heater surface is

purely by single-phase natural convection. Superheated liquid rises to the surface of

the reservoir and evaporation takes place at this surface. As the heat flux is

increased beyond the value at 'B' bubbles begin to form on the surface of the heater,

depart from the heater surface and rise through the liquid this process is referred to

as nucleate boiling. At this stage a reduction of heater surface temperature to 'C'

may be observed. Reducing the heat flux would now result in the heat flux

temperature difference relationship following the curve 'C'-'B*'. This type of

phenomenon, for which the relationship between a dependent and independent

variable is different for increasing and decreasing values of the independent variable,

is known as hysterisis.

After the commencement of nucleate boiling further increase in heat flux leads to

increased heater surface temperature to point 'D'. Further increase beyond the value

2.45

at 'D' leads to vapour generation at such a rate that it impedes the flow of liquid

back to the surface and transition boiling occurs between 'D' and 'E'. At 'E' a stable

vapour film forms over the surface of the heater and this has the effect of an

insulating layer on the heater resulting in a rapid increase in temperature from: 'E' to

'F'. The heat flux at 'E' is known as the critical heat flux. The large temperature

increase which occurs if an attempt is made to maintain the heat flux above the level

of the critical heat flux is frequently referred to as burn-out. However, if physical

burn out does not occur it is possible to maintain boiling at point 'F' and then adjust

the heat flux, the heat flux temperature difference relationship will then follow the

line 'G'-'H'. This region on the boiling curve corresponds to the stable film boiling

regime. Reduction of the heat flux below the value at 'G' causes a return to the

nucleate boiling regime at 'G*'.

The factors which influence the shape of the boiling curve for a particular fluid

include: Fluid properties, heated surface characteristics and physical dimensions and

orientation of the heater. The previous history of the system also influences the

behaviour, particularly at low heat flux.

Clearly several relationships, defining both the extent of each region and the

appropriate shape of the curve for that region, would be required to describe the

entire curve. It is the nucleate boiling region, 'C'-'D' which is of greatest importance

in most engineering applications. However, it is clearly important that the designer

ensures that the critical heat flux is not inadvertantly exceeded, and there are some

systems which operate in the film boiling regime. Many correlations describing each

region of the boiling curve have been published. Additionally, the temperature

difference at which nucleation first occurs, i.e. the temperature at 'C' influences the

boiling regime during flow boiling and the hysterisis.

If the temperature of the heater, rather than the heat flux, was to be controlled then

increasing temperature above that corresponding to the critical heat flux would

result in a decrease in heat flux with increasing temperature from ‘E’ to ‘G’, followed

by an increase along the line G-H. The point ‘G’ is sometimes referred to as the

Liedenfrost Point. Temperature controlled heating of a surface is found in many heat

exchangers and boilers - the temperature of the wall being necessarily below the

2.46

temperature of the other fluid in the heat exchanger. Experimentally, it is difficult to

maintain surface temperatures over a wide range with the corresponding range of

heat fluxes. To obtain boiling curves for varying ∆Tsat it is usual to plunge an ingot of

high conductivity material into a bath of the relevant fluid. The surface temperature

is measured directly and the heat flux can then be calculated from the geometry of

the ingot and the rate of change of temperature.

The explanation for the importance of surface finish lies in the mechanism of bubble

formation. Observation of boiling is difficult because of the vigour with which the

process occurs, high speed photographic or video techniques are necessary to get

anything more than an approximate qualitative overview. However, even this can

give us some insight into the process. Observation of the formation of bubbles in a

carbonated drink in a glass can also be instructive, the following experiment works

best with carbonated mineral water but other drinks can be used. Pour the drink

onto a glass and observe the bubbles. You will note that, once any initial “froth” has

dispersed:

i. Bubbles are formed at the surface of the glass3

ii. Bubbles rise in a chain originating from the same point on the surface.

iii. If the glass is emptied and refilled many of the sites where bubbles form will

correspond to those observed during the first attempt.

This suggests that some feature of the surface encourages bubble nucleation. It has

been observed that nucleation occurs in cavities within the surface, these cavities

contain minute bubbles of trapped gas or vapour which act as starting points for

bubble growth. This is illustrated schematically in Fig. 2.20. When the bubble leaves

the site a small bubble remains in the cavity which acts as the start for the next

bubble.

3 Any bubbles which arise from a point within the bulk of the liquid almost certainly originate at a solid impurity, for example dust or a particle of organic matter.

2.47

Liquid

Surface

Trapped bubblesof gas or vapour

Figure 2.20 Schematic representation of surface showing nucleation sites.

Consideration of idealised nucleation sites allows some indication of their necessary

size if they are to play a part in boiling. With reference to an idealised conical cavity

as shown in Fig. 2.21.

Liquid GrowingBubble

2R

Figure 2.21 Idealised cavity acting as a nucleation site

The pressure, pB, inside a bubble is somewhat higher than the pressure in the

surrounding liquid:

p prB = +

2σ (2.86)

Where p is the liquid pressure, r is the radius of curvature of the bubble and σ is the

surface tension of the liquid. The radius of curvature is a maximum when the bubble

forms a hemispherical cap over the cavity, i.e. r=R, the radius of the mouth of the

2.48

cavity. This is the condition for pB to be a maximum. If the bubble is to grow then

the wall temperature must be sufficiently high to vapourise the liquid at a pressure pB.

In order for the bubble to grow:

(T T d )Tdp

p pW sat B> + − (2.87)

The Clausius-Clapeyron Equation states that the slope of the vapour pressure curve

is given by:

( )dpdT

hv v T

fg

g f sat

=−

(2.88)

if vg is very much greater than vf we can simplify this:

dTdp

v Thg sat

fg

= (2.89)

Hence, for the bubble to grow:

T TRp

v ThW satg sat

fg

> +2σ

(2.90)

The radius of the cavity and the superheat, ∆T sat , at which nucleation from the

cavity starts can be related:

Rv T

h Tg sat

fg sat

=2σ∆

(2.91)

For water boiling at 1bar is commonly of the order of 5K. Substitution of

values for the properties of water gives a value for the smallest active cavity to be

approximately 6.5 x 10

∆T sat

-6m radius. This demonstrates that typical active cavities are of

the order of 1-10µm.

Clearly, real surfaces have a range of cavities of varying size and shape. Surfaces

which are designed to improve boiling heat transfer (enhanced surfaces) are made to

have large numbers of suitable cavities.

2.49

Some useful pool boiling correlations

The symbols used are:

α Heat transfer coefficient W/m2K

σ Surface tension N/m

ρf Liquid density kg/m3

µf Liquid viscosity Ns/m2

ρg Vapour density kg/m3

∆Tsat Temperature difference K

cpf Specific heat capacity of liquid J/kgK

cpg Specific heat capacity of vapour J/kgK

g acceleration due to gravity m/s2

Gr Grashof Number

hfg Latent heat J/kg

kf Liquid thermal conductivity W/mK

kg Vapour thermal conductivity W/mK

Nuf Nusselt Number (liquid conductivity)

Nug Nusselt Number (vapour conductivity)

Pr Prandtl Number

q Heat flux W/m2

It can be argued that the heat transfer coefficient, defined by α = &q Tsat∆ , is of

limited use when it is not constant, but varies with heat flux (or temperature

difference). However many correlations are given in terms of heat transfer

coefficient.

Natural Convection Region:

Typically:

(2.92) Nu CGr Prm m=

Where C and m depend on the geometry and whether the induced flow is laminar

or turbulent

Nucleate Boiling Region:

There are a wide number of correlations which have been applied to nucleate pool

boiling. Some of the more commonly used are given below:

Rohsenow (1952)

This is essentially an empirical correlation, but it is instructive to see the way in

which it was derived.

It is evident that it will be difficult, if not impossible, to produce a theoretical model

of boiling which can be used to predict heat transfer coefficients. The situation is

2.50

complicated by the dependence of the heat transfer on the condition and history of

the surface.

It has already been noted that experimental results for nucleate boiling may be

represented by an equation of the form:

( )

msat

msatwall

Taq

TTaq

∆=

−=

&

&

or (2.93(a))

This may be rearranged in terms of a heat transfer coefficient, α,

nm

m

msat

sat

qbqb

TaTq

&&

&

≡=

∆=∆

=

−

−

1

1

α

α

(3.93 (b)) (3.93 (c))

the value of m is generally in the range 3 - 2.33, corresponding to n being in the

range 0.67- 0.7.

An early nucleate boiling correlation is that due to Rohsenow, following the example

of turbulent forced convective heat transfer correlations Rohsenow argued that:

Nu= f(Re,Pr)

Nu Lk

Re UL

Prck

p

=

=

α

ρµ

µ

(U = Velocity)

=

If the fluid properties are all those for the liquid this still left the problem of choosing

a suitable velocity and representative length, L.

The velocity may be taken as the velocity with which the liquid flows towards the

surface to replace that which has been vapourised:

fgf hqU

ρ&

= (2.94)

and the representative length is given by:

( )5.0

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−=

gfgL

ρρσ

(2.95)

2.51

The correlation thus produced was:

NuC

Re Prsf

x= − −1 1 y (2.96)

Which is frequently presented in the form:

( )y

f

pff

x

gffgfsf

fg

satpf

kc

ghqC

hTc

+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

∆15.0

µρρ

σµ&

(2.97(a))

For most fluids the recommended values of the exponents were: x=0.33, y=0.7.

This correlation then corresponds to:

[ ]q T= Constant depending upon fluid properties and surface x ∆ 3

It may also be written:

( )n

gffgfsf

fg

satpf Prgh

qCh

Tc333.05.0

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

∆

ρρσ

µ&

(2.97(b))

or

( )

&

.

q hg c T

C h Prf fgf g pf sat

sf fgn=

−⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟µ

ρ ρ

σ

0 5 3∆

(2.97(c))

The value of the constant Csf depends upon the fluid and the surface and typical

values range between 0.0025 and 0.015. Since, for a given value of ∆Tsat the heat flux

is proportional to Csf3 the correlation is very sensitive to selection of the correct

value.

It is arguable that the complexity of the correlation is not warranted because of the

need for this factor.

2.52

Some values of Csf for use in Equation 2.97 and are given in the table below.

Fluid Surface Csf

Water Nickel 0.006

Water Platinum 0.013

Water Copper 0.013

Water Brass 0.006

Carbon Tetrachloride Copper 0.013

Benzene Chromium 0.010

n-Penthane Chromium 0.015

Ethanol Chromium 0.0027

Isopropanol Copper 0.0025

n-Butanol Copper 0.0030

Forster and Zuber (1955)

αρ

σ µ ρ=

⎛

⎝⎜⎜

⎞

⎠⎟⎟0 0122

0 79 0 45 0 49

0 5 0 29 0 24 0 240 25 0 75.

. . .

. . . ..

k ch

T pf pf f

f fg gsat sat∆ ∆ . (2.98)

( )∆∆

ph T

Tsat

fg sat

sat g f

=−1 1ρ ρ

(2.99)

Mostinski (1963)

(2.100) (α = + +0106 18 4 100 69 0 17 1 2 10 0 7. . &. . .p p p p qcr r r r ) .

Cooper (1980)

( ) ( )α ε= −− − −55 0 12 0 2 0 55 0 5 0 67p p Mr r. . log . . .log &q (2.101(a))

( )α = −− −55 0 12 0 55 0 5 0 67p p Mr r

. . . .log &q (2.101(b))

ε is the surface roughness in microns. Typically a value of 1 may be used, thus

simplifying the equation.

Mostinski and Cooper are both dimensional equations, therefore the units must be

consistent with the constants given. For the forms quoted here pressures are in bar

and heat flux in W/m2, giving heat transfer coefficients in W/m2K.

2.53

Critical Heat Flux:

Kutateladze (1963) & Lienhard et al (1970,1973)

( )( )CHF q C h gg fg f g= = −&max.

.ρ σ ρ ρ0 5

0 25

For Plates C is in the range 0.13 to 0.18 depending on geometry

For Cylinders

( )( )

( )5.0

44.3exp27.289.013.0

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

−+=

vfb

b

gL

LRC

ρρσ

Film Boiling Region

For spheres and cylinders

( )

( )Nu cgh d

k Tv

v f g fg

g g sat

=−⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

ρ ρ ρ

µ

*.

30 25

∆ (2.103)

c=0.62 for cylinders and 0.67 for spheres. Liquid properties are evaluated at the

saturation temperature and the vapour properties at the average of the surface

temperature, Ts, and the liquid saturation temperature, Tsat. The corrected enthalpy

of vapourisation is calculated from:

h h (2.104) c Tfg fg pg sat* .= + 0 4 ∆

This correction accounts for the sensible heating of the vapour.

For large horizontal surfaces the expression:

( )

( )Nugh d

k Tvv f g fg rep

g g sat

=−⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

0 4253

0 25

.*

.ρ ρ ρ

µ ∆ (2.105)

may be used, where the Nusselt Number is based on the representative dimension

drep, defined by:

( )dg

rep

f g

=−

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

σ

ρ ρ

0 5.

(2.106)

2.54

The situation is further complicated in film boiling because of the high surface

temperatures which may be involved. The radiative heat transferred can be

calculated from:

( )4sat

4srSBrad TTq −= εσ& (2.107)

where is the Stefan Boltzman constant and SBσ 429 km/W107.56 −×= rε is the

emmissivity of the surface.

The convective component is calculated separately. To account for the interaction

between the two mechanisms they should be combined:

& . &q q qcon rad= + 0 75 (2.108)

Film boiling is rarely encountered in heat exchangers, the designer usually wants to

ensure that there is no risk of exceeding the internal hear flux within the exchanger.

This is particularly important when reduction in the heat transfer results in an

increase in temperature of the heating medium. Exceeding the critical heat flux in

these circumstances results in a rapid temperature increase, or burnout.

2.4.3 Flow Boiling

The prediction of heat transfer coefficients in flow boiling is even more difficult (and

often less reliable!) than in pool boiling. In addition to the influence of heat flux (or

temperature difference), fluid and surface properties and geometry we must also

consider the flow velocity and the quality of the fluid.

2.55

Fig 2.22 Regions of convective boiling

We shall firstly discuss boiling in vertical tubes, similar considerations apply to

horizontal tubes.

Let us first consider the flow patterns and boiling regimes in a vertical tube heated

uniformly along its length. This is illustrated in Fig.2.22.

As we proceed up the tube we would observe:

• Region A single phase-convection

• Region B Sub-cooled boiling- as the fluid approaches its saturation temperature the

wall and the fluid adjacent to the wall will exceed the saturation temperature and

nucleation may commence. Bubbles will form and then collapse as they move into

the cooler bulk fluid. (This phenomenon can occur during pool boiling and is

responsible for the “singing” of a kettle prior to boiling)

• Regions C and D are the saturated nucleate boiling regions

• Regions E and F are the convective boiling regions

• Region G is the liquid deficient region

2.56

• Region F Involves single-phase convection to the vapour.

The flow patterns which one observes may be described:

• Bubbly flow The gas phase is present as discrete bubbles dispersed throughout the

liquid phase.

• Slug flow Gas bubbles approaching the diameter of the pipe move up the pipe,

separated from the wall by a descending liquid film. The gas bubbles have

approximately spherical caps (in round tubes). The bubbles are separated by slugs of

liquid, which may contain entrained gas bubbles.

• Churn Flow Long bubbles formed as slug flow develops become unstable and the

gas bubbles and liquid slugs become intermingled. The liquid tends to be displaced

towards the tube wall but intermittent, irregularly shaped liquid bridges pass up the

tube.

• Wispy Annular At high mass velocities the majority of the liquid flow is attached to

the duct walls but "fingers" of liquid flow in the gas core.

• Annular Flow The liquid phase flows principally as a film on the pipe wall while the

gas flows up the central core. Waves forming on the film may break up causing liquid

to be entrained in the gas core as discrete droplets.

• Drop Flow Since the presence of the heated wall causes the liquid film to evaporate

the wall will dry out prior to the thermodynamic quality of the fluid reaching unity.

Drops, entrained in the vapour during annular flow remain in the vapour stream,

only evaporating when the bulk vapour temperature is increased to a value slightly

above the local saturation temperature.

Flow in horizontal channels yields similar patterns, but the effects of gravity result in

stratification, particularly at low velocities. The resulting patterns are shown

schematically in Figure 2.23.

2.57

Figure 2.23 Flow patterns during boiling in a horixontal tube

As you would expect relationships are required for each of the flow regimes and

heat transfer regions. We will deal only with the annular flow regime which occurs

for vapour quality in excess of a few percent and is therefore the most prevalent

regime in practice. For example, in refrigeration evaporators which receive a vapour

liquid mixture from the expansion valve the flow will be entirely annular.

2.58

In

fact the correlations which we will examine can be used with reasonable accuracy

for the complete range of saturated boiling. All flow boiling correlations are

empirical, but are based upon observations of the mechanisms involved as well as

heat transfer data. Heat transfer in flow boiling can be regarded as being due to one

or both of two mechanisms: namely nucleate boiling and convective boiling. In

general low quality and high heat flux favour nucleate boiling while high quality and

low heat flux lead to convective boiling. High mass flux is conducive to convective

boiling. Figure 2.24 illustrates the nucleate and convective boiling regions.

Figure 2.24 Variation in heat transfer coefficient Figure with quality, heat flux and mass flux

.

The way in which correlations account for the two mechanisms differ, some add the

contributions for each mechanism, some take only one contribution, and some

combine the contributions so that the effect of the larger is dominant. The general

form of these correlations is illustrated below and in figure 2.25. It must be

remembered that very high heat fluxes can lead to formation of a vapour film,

analogous to that encountered in pool boiling. The analysis presented here assumes

that the heat fluxes encountered are molecular.

2.59

Figure 2.25 Schematic illustration of Correlations

Additive or superposition (e.g. Chen, (1963)):

Lconv

nbpnb

convnb

F

S

α=α

α=αα+α=α

(2.109)

Where the subscripts have the following meanings

nb nucleate boiling contribution

nbp predicted for pool boiling at the same temperature difference from Forster

and Zuber according to Chen)

conv convective boiling contribution

l predicted for the single phase flow of liquid (either all fluid flowing as liquid

or based on the liquid component only) (for Chen this is from Dittus Boelter with

liquid only Reynolds No.)

S and F are factors which are correlated against flow parameters

Enhancement or Substitution (e.g. Shah (1976) )

α α= E l

E is an enhancement factor, the value of which is given by one of several expressions

depending upon the flow parameters and heat flux.

2.60

Asymtotic (e.g. Liu and Winterton (1988))

α α αα α

α α

2 2 2= +=

=

nb conv

nb nbp

conv l

SF

(2.110 )

The factors in the Liu-Winterton correlation may be determined from:

F xPr

SF Re

ll

g

L

= + −⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

=+

1 1

11 0 055

0 35

0 1 0 16

ρρ

.

. ..

(3.110)

( )α

α

µ

nbp r r

LL

L L

LL

total

p p M qkd

Re Pr

ReGd

Gm

Flow Area

= −

=

= =

− −55

0 023

0 12 0 55 0 5 0 67

0 8 0 4

. . . .

. .

log &

.

,&

Examination of Figs 2.22 and 2.25 and the form of typical boiling correlations shows

that the heat transfer coefficient during flow boiling varies significantly as the quality

goes from 0 (pure liquid) to 1 (dry vapour). If the vapour is then superheated there

will be a step change in heat transfer coefficient as the wall dries out. This means

that a stepwise approach must be taken in the design of flow boilers: The local film

heat transfer coefficients and overall heat transfer coefficient must be evaluated at

entry to the channel, the heat transferred over a short length of channel evaluated

thus permitting calculation of the increase in quality over the short length. This

process must be repeated over the length of the tube to determine the total heat

transferred. Clearly this is a very time consuming process and best carried out using

a computer package.

Correlations are available which can be used to give estimates of the mean heat

transfer over a range of vapour qualities, for example the Pierre (1964) correlation:

2.61

( ) n

fginoutL

l hxxdk

C ⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆−=

Length TubeRe2α (2.111)

where C=0.0009 and n=0.5 for exit vapour quality up to 0.9 and C=0.0082

and n=0.4 for higher vapour qualities and exit superheat of up to 6K.

It should also be noted that the concepts of mean temperature difference and

effectiveness covered in Section 5 rely upon an assumption that the heat transfer

coefficient is constant over the entire heat exchanger area.

In at least the preliminary stages of thermal design it may be permissible to use an

average heat transfer coefficient either for the whole heat exchanger, or for

particular sections. For example, if subcooled liquid enters and this is fully

evaporated and then superheated the heat exchanger may be considered in three

sections - the economiser, the boiling section and a superheater.

Finally, in many applications involving boiling, for example fired boilers, the boiling

side heat transfer coefficient is likely to be very much higher than the heat transfer

coefficient from the heating medium to the wall, hence variations in the boiling side

heat transfer coefficient have little influence on the overall heat transfer coefficient.

2.62

2.4.4 Condensation

Condensation involves the formation of a liquid from a vapour due to heat transfer

from the fluid or a change in pressure of the fluid. The various modes of

condensation which may be observed are illustrated in Figure 2.26.

Figure 2.26 Modes of Condensation

2.63

Modes of condensation

• Filmwise condensation: The condensate forms a continuous film on the cooled

surface. This is the most important mode of condensation occurring in industrial

equipment and is discussed further below.

• Homogeneous condensation: The vapour condenses out as droplets suspended in

the gas phase, thus forming a fog. A necessary condition for this to occur is that the

vapour is below saturation temperature, which may be achieved (as illustrated) by

increasing the pressure as the vapour flows through a smooth expansion in flow

area. In condensers, however, it usually occurs when condensing high-molecular-

weight vapours in the presence of noncondensable gas. Fogs may also form when

cold gas is mixed with vapour, for example, during the mixing warm, humid air with

cold air.

• Dropwise condensation: This occurs when the condensate is formed as droplets

on a cooled surface instead of as a continuous film. High heat transfer coefficients

can be obtained with dropwise condensation, but this is difficult to maintain

continuously In heat exchangers.

• Direct contact condensation: This occurs where vapour is brought directly into

contact with a cold liquid.

• Condensation of vapour mixtures forming immiscible liquids: A typical example of

this is when a steam-hydrocarbon mixture is condensed. The pattern; formed by the

liquid phases are complicated and varied

Filmwise Condensation:

Filmwise condensation occurs when the condensate vapour forms a film on the

surface which runs down the surface, as shown in Fig. 2.27. The film will be laminar

(and amenable to analysis) at the top of the surface, as the film becomes thicker the

laminar flow is not stable and waves form in the film, lower down the surface the

film becomes turbulent. In many heat exchanger applications, it is satisfactory to

assume laminar flow. This gives a conservative estimate of the heat transfer, since

both waves and turbulence lead to an increase in the heat transfer coefficient.

2.64

Fig. 2.27 Schematic representation of filmwise condensation on a vertical plate

Rogers and Mayhew (1980) present the analysis of filmwise condensation in the

laminar non-wavy region originally derived by Nusselt. This analysis is summarised

below.

The heat ransfer coefficient a distance x from the top of the plate may be calculated

from:

25.0

satff

32f

'fg

x Tk4

gxhNu ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛=

∆µ

ρ (2.112)

The mean heat transfer coefficient from the top of the plate to some point l below

the top may be calculated from:

Nuh gl

k Tfg f

f f sat=

⎛

⎝⎜⎜

⎞

⎠⎟⎟

43 4

2 3 0 25' .ρ

µ ∆

2.65

and, for a horizontal tube, diameter d:

Nuh gd

k Tdfg f

f f sat=

⎛

⎝⎜⎜

⎞

⎠⎟⎟103

4

2 3 0 25

.' .ρ

µ ∆

( in the above equations) (∆T T Tsat sat w= − )

Properties are evaluated at the arithmetic mean film temperature, with the

exception of the latent heat which should be calculated from:

h h c Tfg fg pf sat' .= + 0 68 ∆

with hfg calculated at the saturation temperature.

Corrections are available to take account of:

Waves and Turbulence

Shear between the liquid and vapour

In the absence of the above, the Nusselt Equation for horizontal tubes may also be

used for tubes in the bundles of heat exchangers. However, condensate will drain

from the upper tubes to the lower tubes thus increasing the film thickness on all but

the top tube.

If the liquid flows uniformly then the mean heat transfer coefficient for a bank N

rows deep is given by:

αα

N N1

0 25= − .

and for a tube on the Nth row the heat transfer coefficient is given by:

( )αα

N N N1

0 75 0 751= − −. .

2.66

In fact, in most practical situations the liquid flows to the lower tubes in rivulets as

shown in (b), and the reduction in heat transfer coefficient is not as marked as

predicted by the above equations.

Filmwise condensation - Nusselt analysis

Filmwise condensation on a vertical surface is one of the few aspects of convective

heat transfer, which yields to an analytical solution. Nusselt derived a solution based

on the following assumptions:

• The shear force between the vapour and the condensate film is negligible

• Inertia ad hydrostatic forces in the film may be neglected

• The flow of liquid in he film is laminar

• The resistance to heat and mass transfer at the liquid vapour interface is

negligible

With reference to figure2.28

If the thermal conductivity of the film, thickness δ, is constant then:

( )dQ kt t

kdx t t dx tw s

x w s x& = −

−= − − = −α α ∆ s (2.117)

where:

αδx

k=

Defining the Nusselt Number, a distance x from the top of the film as:

Nuxxxk

x= =α

δ

suggests that to find the heat transfer coefficient we must first find the film thickness

δ at a distance x from the top of the film.

2.67

(b) (a)

Figure 2.28: Schematic representation of filmwise condensation

Now, if we consider an element of the film, length dx and an element of the fluid

shaded in figure 2.26 , we can equate the shear force and gravity acting on the

element:

( )τ µ δ ρdxdUdy

dx y dx g=⎛⎝⎜

⎞⎠⎟ = − (2.118)

or

( )dUg

y dy= −ρµ

δ (2.119)

This can be integrated to find the velocity distribution through the film:

Ug

yy

= −⎛⎝⎜

⎞⎠⎟

ρµ

δ2

2 (2.120)

The mass of liquid flowing through the film at some distance x from the top is then

given by:

µδρδ

µρρ

δδ

3gdy

2yygUdym

32

0

22

0

=⎟⎟⎠

⎞⎜⎜⎝

⎛−== ∫∫& (2.121)

2.68

At a distance dx below, when the film thickness has increased by dδ, we can write:

( )& & & &m dm m dmg

d+ − = =ρ δµ

δ2 2

(2.122)

Now the heat transferred may be related to the latent heat given up by the

condensing vapour:

dQ h dm hg

dfg fg& &= =

ρ δµ

δ2 2

(2.123)

Giving:

dQ kdxt

hg

dsfg

& = =∆δ

ρ δµ

δ2 2

(2.124)

Integrating between x=0, where δ= 0, and x, the film thickness at x may be

determined:

δµρ

42

4=

k t xh g

s

fg

∆ (2.125)

or:

25.0

satff

32f

'fg

x Tk4gxh

Nu⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

∆µρ

(2.112)

If the plate is inclined to the vertical at an angle β to the vertical then we may

substitute g cosβ for g.

The mean heat transfer coefficient from x=0 to x=l is given by:

l

l

0lx 3

4α34dxα

l1α NuuNor === ∫ . (2.126)

For a cylinder, diameter d, the effective length or equivalent plate height is l=2.85d.

Any aspect of the geometry or fluid flow which causes the film to break up or

become wavy tends to enhance condensation heat transfer, as does shear due to

high velocity vapour.

2.69

The presence of non-condensable gases, even in small quantities, in a condenser can

have highly detrimental effects on the condenser performance. The non-condensable

gas (usually air) becomes concentrated adjacent to the liquid film, thus forming a

layer through which vapour must diffuse. The partial pressure of the vapour, and

hence its condensing temperature, is reduced by the presence of non-condensables

therefore the temperature at the surface of the film, and consequently the

temperature difference across the film, is reduced.

2.70

2.5 Fouling of Heat Exchangers

Fouling of Mechanisms

The deposition of foreign matter on a heat transfer surface is known as fouling. The

presence of a foulant on a surface introduces an additional thermal resistance

between the surface and the heat transfer fluid. As the layer becomes thicker this

effect becomes more marked and, since the foulant occupies space within the flow

passage, effective diameter of the flow passage decreases with a consequential

increase in pressure drop (or reduction in flow rate). Both of these effects are

undesirable and therefore heat transfer equipment and process conditions must be

designed to minimise the effects of fouling. Measures to mitigate the effects of

fouling may be preventative (eg treatment of cooling water, high fluid flow velocities),

or remedial (eg regular cleaning of the affected surfaces). Additionally, it is usual to

allow for a thermal resistance due to fouling when specifying or designing a heat

exchanger.

Unfortunately, the complex mechanisms involved in fouling are not fully understood

and there is only a limited theoretical background to permit the fouling propensity of

new designs or applications to be predicted. In practice a designer must rely upon

the TEMA fouling factors4 which is additional thermal resistance’s which should be

incorporated into the determination of the overall heat transfer coefficient when

designing a shell-and-tube heat exchanger.

For convenience fouling is generally classified under one of six headings depending

upon the mechanism causing the deposition eg5:

a) Crystallisation or precipitation fouling occurs when a solute in

the fluid stream is precipitates and crystals are formed either on the heat transfer

surface or in the fluid and subsequently deposited on the heat transfer surface.

When the fluid concerned is water and calcium or magnesium, salts are deposited.

This mechanism is frequently referred to as scaling. 4 Tubular Exchanger Manufacturers Association 5 Bott T.R., General Fouling problems, Fouling Science and Technology, NATO ASI Series, Ed., Melo L.F., Bott T.R. and Bernado C.A., Kluwer Academic Publishers 1988.

2.71

b) Particulate fouling (silting) occurs when solid particles from the

fluid stream are deposited on the heat transfer surface. Most streams contain some

particulate matter originating from a variety of sources.

c) Biological fouling is caused by the deposition and growth of

organisms on the heat transfer surface.

d) Corrosion fouling is the result of a chemical reaction involving the

heat transfer surface leading to a build up of corrosion products on the surface.

e) Chemical reaction fouling occurs when a reaction involving one

or more constituents in the process fluid results in the formation of a solid layer on

the heat transfer surface. The surface itself is not involved in the chemical reaction.

f) Freezing or solidification fouling occurs when the temperature of

the process fluid is reduced sufficiently to cause freezing at the heat transfer surface.

The above definitions are commonly used. However it must be noted that other

classification are also found in the literature and in specialist publications. For

example, defines scale, microbiological contamination and corrosion corresponding

broadly with (a), (c) and (d) above but reserves the term fouling for deposition of

particulate matter, as in (b) above.

Fouling of Open Cooling Water Systems

In open cooling water systems neither chemical reaction nor freezing (e) nor (f)

above is likely to occur (freezing of water in the cooling tower pond or connecting

pipe work during cold weather is a separate problem). System design, materials

selection and water treatment combine to mitigate the effect of scaling, corrosion

and particulate and biological fouling, however one or more of these mechanisms

causes some degree of fouling in most practical open cooling systems. It should also

be emphasised that the mechanisms described do not operate independently of each

other but usually occur concurrently and can interact.

Use of a biocide together with maintenance of the cooling tower to prevent the

establishment and build up of any biological growth can minimise, if not entirely

eliminate, biological fouling. Biocide treatment of the water in cooling tower

systems is essential to eliminate the build up of bacteria which may be harmful to

2.72

health, the best known of these being the legionella bacteria. The materials of

construction of heat exchangers used in cooling water circuits should be chosen so

that corrosion is acceptable, remembering that any corrosion products may act as a

foulant where formed or break away and contribute to particulate fouling elsewhere.

Calcium and magnesium compounds (carbonates, sulphates and phosphates) are

inverse solubility salts, that is there solubility decreases with increasing temperature.

These salts are the principal components of scale in open water systems. Water

treatment must be employed to prevent (or at least minimise) scaling. An adequate

purge rate should prevent unacceptable concentrations of the salts likely to

crystalise, while chemical additives increase the solubility of the common hardness

salts. Lowering the pH of the cooling water increases the solubility of the scale

forming constituents, but tends to raise the potential for corrosion. Dispersants are

chemicals which impart electrical charges to the heat transfer surfaces and particles

so as to keep the particles in suspension.

Cooling water treatment is a specialised field and in designing or operating cooling

water plant it is usual to consult a chemical supplier for advice on the use of

additives.

An additional problem associated with compact heat exchanges and related to, but

not normally classified as, fouling must be considered: It is inevitable that the small

flow passages inherent in most forms of compact heat exchanger will be susceptible

to blockage or plugging by large or fibrous particles. Therefore process fluids for

use in PCHE’s must be filtered to ensure that particles of dimensions comparable to

or larger than the passage cross-section do not reach the heat exchanger. This may

require the use of special filtration equipment and/or more rigorous maintenance

then would be normal for shell-and-tube units. For the purposes of this report the

term ‘blockage’ is used to describe the obstruction of flow passages by relatively

large particles.

2.73

Fouling Rate

It is beyond the scope of this course to evaluate the various models which have been

proposed to facilitate the prediction of fouling behaviour under various conditions

but it is necessary to enumerate some of the factors which influence fouling.

Fouling involves the deposition of material onto the heat transfer surface occurring

concurrently with removal of material previously deposited. A simple model, due to

Kern and Seaton, expresses this:

fW21 xaucafdtdx τ−= (2.127)

Where the first term on the right hand side represents the rate of deposition on the

surface and the second term represents the removal rate.

Integration of equation (2.127) gives:

)(( )Btexp1xx *fF −−= (2.128)

Implying that the fouling thickness approaches a value asymptotically. The values

of and B are given by:

*fx

*fx

w2

1*f a

ucaxτ

= (2.129)

w2aB τ= (2.130)

Equation 2.129 may be derived from examination of equation 2.127: is the value

of the thickness of the fouling layer at which the rate of deposition onto the surface

is equal to the rate of removal.

*fx

2.74

Figure 2.29 Typical fouling curves This

model is not universally applicable. Several curves of fouling resistance (or thickness)

against time have been observed and some typical shapes are shown in Fig. 2.29, the

Kern-Seaton model applies only to curve B, representing an asymptotic deposit with

no induction period. However, this simple model is adequate as a qualitative

indication of the importance of various parameters in determining the rate and

severity of fouling.

If the fouling follows a curve of the form B or D then the heat exchanger can be

designed for continuous operation with a fouling factor corresponding to the

resistance of the layer at the asymptotic thickness. If the foulant continues to build

up then a permissible resistance should be included in the design and cleaning

scheduled to take place before this level is reached. The existence of an induction

period in many situations may lull the operator into a false sense of security, fouling

2.75

is not immediately apparent but appears after an extended period of operation.

There are two possible explanations for this phenomenon.

• Foulant may not initially adhere to the heat transfer surface and the

layer does not build up until the surface has become conditioned in some way.

• Alternatively, if the layer thickness is inferred from heat transfer

measurements then the existence of a fouling resistance may be masked by an

enhancement of the heat transfer coefficient by roughening of the surface. Indeed, a

net increase in heat transfer (corresponding to an apparently negative heat transfer

coefficient) is sometimes observed during the early stages of a heat exchangers life.

Remembering that the wall shear stress, wτ , increases with mean velocity, u, to a

power greater that l, we can see from equations 2-4 that the fouling rate and final

thickness of the fouling layer can be expected to increase with decreasing velocity.

For this reason it is essential that heat exchanger designers avoid regions of low

velocity in their designs. Designers must also be wary of including too conservative a

fouling factor - it may be self fulfilling. If the incorporation of additional heat transfer

area is accompanied by an increase in flow area and corresponding reduction in fluid

velocity (which in itself will reduce the film heat transfer coefficient), then the

propensity to foul will be greater.

In general the higher the temperature of a surface the greater its propensity to foul.

This is clearly the case for deposition of inverse solubility salts or the products of

decomposition. There are obvious exceptions, for example freezing occurs at low

temperatures as does the condensation of liquids or tars from a gas stream (eg

combustion products). The designer should attempt to ensure a uniform

temperature where possible in a heat exchanger.

Tema Fouling Factors

It is often the case that the best that a designer can do is to incorporate TEMA

fouling factors into the evaluation of heat exchanger overall heat transfer coefficient.

These fouling factors have many shortcomings: they take little account of fluid

2.76

velocity or temperature, they apply only to tubular exchangers and to a limited range

of fluids.

Typical values of fouling factors are given in Table 2.7.

2.77

Table 2.7 Typical film transfer coefficients for shell -and-tube heat exchangers (taken from Handbook of Heat Exchangers Design by G.F. Hewitt)

2.78

Table 1.2 Typical film transfer coefficients for shell -and-tube heat exchangers (taken from Handbook of Heat Exchangers Design by G.F. Hewitt)

2.79

Table 2.7 continued. Typical film transfer coefficients for shell -and-tube heat exchangers (taken from Handbook of Heat Exchangers Design by G.F. Hewitt)

2.80

2.5.1 Example Showing Effect of Fouling

a) What features of gasketed plate heat exchangers make them attractive for use for

processing foodstuffs.

b) The figure below shows a pasteurisation system treating 7600 l/hour of milk. It

incorporates two gasketed plate heat exchangers. The regenerator has 51 thin plates

clamped between end plates. The channels between the plates (including those

between the end plates and heat exchanger plates) may be regarded as rectangular,

having width 300mm and the spacing between the plates is 1mm.

The pasteurisation process requires that the milk leaving the heater and returning to

the regenerator is always at 77oC. When the plates of the regenerator are clean the

milk enters the heater at 65oC. Calculate the height of the plates in the regenerator.

After a period of operation, fouling of the plates occurs and a fouling resistance of

0.0001m2K/W is applied to each surface. Estimate the percentage increase in the rate

of energy supplied to the heater to maintain the milk peak temperature at 77oC.

Calculate the temperature of the milk leaving the plant.

For the plate heat exchanger the heat transfer coefficient may be calculated from:

Nu = 0 2536 0 65 0 4. Re Pr. .

Properties of milk:

Density 1030kg/m3

Specific heat capacity 3.92kJ/kgK

Dynamic viscosity 1100 x 10-6kg/ms

Thermal conductivity 0.565W/mK

2.81

Regenerator

Heater

77oC

Pasteurised milk

Milk from storage at

4oC

Solution Showing Effect of Fouling

Plate Heat Exchanger

51 plates + ends therefore 52 channels

26 channels per side.

Flow area/channel

mm2602

3004dh ××

= 2610300

mAmG 6 ××

== −&&

Flow sec/111.2sec/36007600hour/7600 lll ===

sec/kg174.203.1111.2m =×=&

506101100

1022610300

174.2GdhRe 6

36

=×

××⎟⎠⎞

⎜⎝⎛

××== −

−−

µ

63.7565.0

3920101100kC

Pr6

p =××

==−µ

8.32PrRe2536.0Nu 4.065.0 ==

Km/W9254102

565.08.32d

kNu 23

n=

×

×=

×=

−α

Since properties are constant and the same flow in each side:

ααα == 21 for thin plate 0kt≈

2.82

Km/W4627U11U1 2

21=∴==

αα

( ) ( ) ( )( )out,hin,hpin,cout,ccp TTcmTTcm −=− &&

C65T 0out,c = , , C4T 0

in,c = C77T 0min =

( ) ( )npcp cmcm && =

C16)465(77T 0out,n =−−=∴

C12416T 01 =−=∆ C126577T 0

2 =−=∆ C12T 0m =∴∆

( ) kW52061392174.2TTcmQ out,hin,hp =××=−= &

mTUAQ ∆=

23

m36.9124627

10520A =×

×=

After Fouling

km/W2403U

0001.00001.04627

1

rrU1

U1

2f

2f1ff

=

++=

++=

mf TAUQ ∆= Heat Transfer

( ) ( )out,nin,hin,cout,c TTTT −=− Heat Balance

( )in,cout,cp TTcmQ −= & (1)

( ) ( )out,cminin,cout,hm TTTTT −=−=∆ [ Equal ] pCm&

2.83

( )in,cout,cin,hout,h TTTT −−= (2)

( )in,cout,hf TTAUQ −= (3)

ombine (1), (2) and (3)

C

( )in,cout,c

in,cin,cout,cin,h

in,cout,c

in,cout,hp

TTTTTT

TTTT

UAcm

−−−−

=−−

=&

1TTTT

AUcm

in,cout,c

in,cin,h

f

p −−−

=&

Substitute values for

0in,h = 0

in,c 4T = T C77

( ) C/kW92.3174.2cm 0p ×=&

C/kW36.9403.2AU 0f ×=

C57T 0out,c =

Original heater power )6577(cm p −&

)5777(cm p −& Fouled heater power

Percentage waste ncreasein%67

6577)6577()5777(100

=−

−−−×=

C24)457(77T 0out,n =−−=

2.84

2.6 Heat Transfer by Radiation Unless at a temperature of absolute zero (i.e 0.0K or -273.15oC, a situation never encountered in

practice) all matter emits electromagnetic radiation. The higher the temperature of the body the

greater is the rate of energy emission. Bodies also absorb at least a proportion of the thermal

radiation which is incident upon them. Therefore if two bodies which are at different temperatures

are placed so that each intercepts radiation from the other then there will be a net interchange of

energy from the hotter to the cooler body. This is commonly referred to as heat transfer by radiation

or radiative heat transfer.

Electromagnetic radiation requires no medium for its propogation and will therefore pass through a

vacuum. Electromagnetic radiation at the frequencies which are of interest for heat transfer (thermal

radiation) will also pass through most gases. For most applications it can be assumed that gases are

transparent to thermal radiation and do not emit thermal radiation. There are, however, some

important exceptions: the influence of the so called “greenhouse gases” in the atmosphere being one

and radiation from flames and combustion products being another.

Before considering the transfer of energy it is necessary to remind ourselves of the nature of the

radiation involved.

2.6.1 The spectrum of electromagnetic radiation.

We do not need to study the physics of electromagnetic waves in any detail, it is sufficient to know

that they are characterised by their frequency or wavelength. Wavelength is inversely proportional to

frequency:

vc=λ

Where:

λ = wavelength

c = the velocity of light

v = frequency

Frequency is expressed in Hertz (1/s) and velocity in m/s therefore, for consistency of units

wavelength is given in metres. However, in descriptive work the wavelength may ne quoted in cm,

mm or µm.

Fig shows the electromagnetic spectrum and the names associated wih various wavelengths of

radiation. The frequency and hence wavelength depend upon the nature of the source. Radiation in

the wavelength range 0.1-1000 µm ( 10-5-10-1cm) will heat any body on which it is incident and is

known as thermal radiation. Thermal radiation encompasses ultra-violet radiation, visible light and

2.85

infer-red radiation. The visible spectrum falls within the wavelength band 0.38-0.76 µm. The quantity

and frequency of the radiation emitted by a body depends upon the temperature of the body, we

cannot see thermal radiation emitted from a body at a temperature below about 500oC.

Thermal Radiation

2.6.2 Black Body Radiation When radiation is incident upon a body it may be reflected, transmitted or absorbed, or a

combination of two or three of these.

Defining the following terms:

τ

α

ρ

==

==

==

EnergyIncident Energy dTransmittevity Transmissi

EnergyIncident Energy AbsorbedtyAbsorbtivi

EnergyIncident Energy Reflectedty Reflectivi

(Note symbols – these are widely used in the literature but are also used in other areas of heat

transfer)

Since the entire incident radiation must be reflected, transmitted or absorbed:

1=++ ταρ (2.131)

For most engineering applications solids are opaque to thermal radiation, i.e. τ=0. Even optically

transparent substances are opaque to all but a narrow range of wavelengths. If τ=0

1=+αρ (2.131(a))

2.86

If all the radiation of all frequencies incident on an object is absorbed then the object is known as a

Black Body. (Note: the terms object or body and surface are almost interchangeable in this context,

almost all absorption occurs within a few microns of the surface of an object)

For a black body 0 and 1 == ρα . The visual appearance of an object or material is not always a

good guide to its “blackness”. For example snow is almost black to thermal radiation outside the

visible range.

It can be demonstrated that a black body is, for a given size and temperature, the best possible

emitter of radiation.

Consider a small object in a large enclosure:

Fig. 2.30 Object in large surroundings

If the body and the enclosure are at the same temperature then there can be no net exchange of

energy by radiation (or by conduction or convection). This is a consequence of the Second Law of

Thermodynamics, but hopefully it is intuitively obvious.

First assume that the object is a black body. All the energy incident on the body is absorbed. For

equilibrium an equal amount of energy must be radiated by the body. Let this amount be , where

A is the surface area of the body. If the black body is then removed and replaced by a body of the

same shape and size but having a surface such that

AEb

1<α then some of the incident radiation must be

reflected from the surface of the body. The amount of radiation incident on the body will be

unchanged since this depends only on the temperature of the surroundings and the dimensions of the

object. The rate of energy incident upon the body is still and the rate of energy absorption is AEb

AEbα . For thermal equilibrium these must be equal to the rate at which energy is radiated from the

body, EA.

b

b

EE

AEEA

=

=

α

α (2.132)

2.87

Since α must be less than one, then E is less than Eb. E is known as the emissive power of the body

and is equal to the energy radiated per unit time per unit surface area of the body. The ratio of the

emissive power of a body to the emissive power of a black body having the same dimensions is known

as the emissivity, ε.

bEE=ε (2.133)

It can be deduced from the above discussion that the emissivity of a body is equal to its absorbtivity at

a given temperature. The emissivity of a body radiating energy at a temperature T is equal to its

absorptivity at the same temperature , T.

The rate at which energy is radiated from a black body may be determined from the Stefan-Boltzmann

law:

4TAEb σ= (2.134)

Where:

Eb = the emissive power W

σ = the Stefan-Boltzmann constant W/m2K4

T = the absolute temperature K

A = the area of the body m2

The Stefan-Boltzman constant has a value 5.67 x 10-8W/m2K4

In practice, no surface is absolutely black to thermal radiation but many surfaces approach the ideal

having emissivities in excess of 0.95.

The surroundings may frequently be considered to behave like a black body. Reference to figure

2.31(a) shows that very little radiation leaving a small object in large surroundings will be reflected

back to the object, therefore the surroundings appear to be black. Similarly a small hole leading to a

relatively large chamber, as shown in figure 2.31(b), will appear black since radiation entering the hole

will not be reflected out. Even if the surfaces in question have high reflectivity, radiation will be

absorbed during multiple reflections.

(a) Small object in surroundings (b) Small hole in chamber

Fig. 2.31Approximations to black body

2.88

When a relatively small object radiates heat to large surroundings at uniform temperature, the net

rate of heat transfer to the body is given by:

)( 441 sTTAQ αεσ −−=& (2.135(a))

which, if ε and α are independent of temperature, can be written:

)( 441 sTTAQ −−= εσ& (2.135(b))

The range of wavelengths of the radiation emitted from a black body, as well as the rate of energy

emission depends upon its temperature. Figure 2.32 shows the variation with temperature of

wavelength in terms of the emissive power/micron of wavelength for a black body.

Fig. 2.32 Spectral distribution from a black body

The distribution is given by the equation:

( ) ⎥⎦

⎤⎢⎣

⎡−

=1exp

125

2

TKhchcE

ob λλ

πλ (2.136)

and the wavelength, maxλ at which the emissive power is a maximum is at a given temperature can be

determined from:

cT =maxλ (2.137)

where the symbols have the following meanings and values.

2.89

A body or surface which emits less than a black body but has the same shape spectral distribution, as

shown in figure 2.33 is said to be grey. For a grey surface ε does not vary with temperature, however

in many instances the error introduced by assuming constant emissivity is acceptable.

Real surface approximating grey body distribution

Grey body distribution

Black body distribution

λE

λ

Fig. 2.33 Spectral distribution for grey body

The emissivities of various surfaces are given in Table 2.8.

Note that for real surfaces the emissivity may vary significantly with temperature. If the body is at T1

and the surroundings are at Ts then when calculating the energy exchange the emmisivity at T1 and the

absorbtivity at Ts should be used. i.e. the net rate of heat transfer to the body is given by:

)( 4411 sTsT TTAQ αεσ −−=& (2.138)

Selective surfaces are those which have very different values of emmissivity ans absorptivity at

different temperatures are known as selective surfaces. They are particularly useful in solar energy

applications. For solar collectors it is desirable to have a surface with high absorptivity for radiation

emanating from a high temperature source and low emissivity at low temperature (the nature of solar

radiation at the earth’s surface is such that the sun may be approximated as a black body having

temperature ~6000K while the collector surface is at ~350K ) .

2.90

Table 2.8 Emisivities of various surfaces

2.6.3 Practical heat transfer calculations

Body in black surroundings We have already seen that the heat transfer between a body and relatively large surroundings is given

by:

)( 4411 sTsT TTAQ αεσ −−=& (2.138)

Radiation exchange between two black surfaces

In general, for any two objects in space, a given object 1 radiates to object 2, and object 2 radiates to object 1 and both radiate to space. This is illustrated for the general case in figure 2.34

2.91

Radiation to space

Radiation to space

Radiative exchange

Fig. 2.34 Radiation between two bodies

A2, T2

Heat transfer Surface 1

A1, T1

Surface 2

Fig. 2.35 Radiation between two arbitrary surfaces

In order to calculate the energy interchange between the two surfaces at different temperatures it is necessary to calculate both the total quantity of radiation leaving each surface and the proportion of the radiation which reaches the other surface. The radiation leaving a black surface is given by equation 2.134.

The proportion of the radiation which is incident on the other surface is given by the radiation shape factor or view factor, F,. With reference to figure 2.35:,

F1-2 = fraction of energy leaving 1 which reaches 2

F2-1 = fraction of energy leaving 2 which reaches 1

F1-2 and F2-1 are functions of geometry only.

2.92

For body 1, we know that is the emissive power of a black body, so the energy leaving body 1 is . The energy leaving body 1 and arriving (and being absorbed since, by definition,

411TAσ

1=α for a black body) at body 2 is . The energy leaving body 2 and being absorbed at body 1 is . The net rate of energy interchange from body 1 to body 2 is:

214

11 −FTAσ

124

22 −FTAσ

21124

22214

11 −−− =− QFTAFTA &σσ (2.139)

Note that if the two bodies are at the same temperature TTT == 21 then there can be no heat

transfer between them:

0124

2214

1 =− −− FTAFTA σσ

hence:

122211 −− = FAFA 2.140

Equation 2.140 is a useful relationship in determining view factors.

View factors may be obtained analytically for simple shapes and resulting relationships are given in

figure 2.36 or graphically as shown in figure 2.37.

2.93

Fig. 2.36 View factors for various geometries (From Fundamentals of heat transfer, F.P.

Incropera and D.P. DeWitt, John Wiley and Sons)

2.94

Fig. 2.37 Graphical representation of view factors

Radiation exchange between two grey surfaces

When dealing with finite grey surfaces it is necessary to consider both the view

factor and the radiation reflected from one body which is returned to the original

2.95

source. The mathematical manipulation becomes rather complex. We will therefore

limit our considerations to the relatively simple case of infinite parallel plates ( or

long concentric cylinders with a small gap between them) this is shown schematically

in figure 2.40.

Multiple reflection of radiation

1111 ,,,, TA ρεα

2222 ,,,, TA ρεα

Fig. 2.40 schematic representation of radiation between infinite grey surfaces

The radiation emitted from surface 1 is given by . Surface 2 absorbs: 411 TA σε

4121

4121 TATA σεεσαε ≡

and reflects:

4121 TA σρε

Surface 1 absorbs a portion of this radiation: 4

1211 TA σρεα while reflecting

41211 TA σρερ

This series can be developed over the multiple reflections, and the net energy leaving

surface 1 and being absorbed by surface 2 is:

( ) ( ) ( )( )................1 321

22121

41211 ρρρρρρσεε +++= ATQ&

Similar logic gives:

( ) ( ) ( )( )................1 321

22121

42212 ρρρρρρσεε +++= ATQ&

and the net rate of heat transfer is:

( ) ( ) ( ) ( ) ( )( ).........1 321

22121

41

422112 ρρρρρρσεε +++−=−= ATTQQQ &&&

The series

( ) ( ) ( )21

321

22121 1

1........1ρρ

ρρρρρρ−

=+++

( )ATTQ 41

42

21

21

1−

−= σ

ρρεε&

which may be rearranged with the substitution ( ) ( )εαρ −=−= 11

2.96

( )ATTQ 41

42

21

1111

−

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

= σ

εε

& (2.141)

This reduces to equation 2.135(a) if one surface is black , and 2.135(b) if both surfaces are black.

Summary Points

• In order to design or analyse the performance of a heat exchanger or

evaluate the heat transfer performance of a system it is necessary to be able

to relate the rate of heat transfer to the temperature difference between the

two fluid streams or between surfaces and the surrounding fluid.

• Heat transfer between two streams occurs by convection from the hot

stream to the wall, by conduction through the wall and then by convection

from the wall to the cool stream.

• The rate of heat transfer is generally expressed:

( )ch TTUAQ −=&

• The overall heat transfer coefficient, U, and the appropriate area, A, may be

calculated from a knowledge of the heat exchanger geometry and the fluid

flow characteristics.

• Convective heat transfer coefficients are frequently empirical or semi-

empirical and it is essential that an appropriate correlations is used.

• Heat transfer may be adversely influenced by fouling which must be

considered at the design stage.

• Techniques for calculating radiative heat transfer are available

2.97